In this paper we extend previous results concerning the behaviour of JSJ decompositions of closed 3-manifolds with respect to the profinite completion to the case of compact 3-manifolds with boundary.
We also illustrate an alternative and perhaps more natural approach to part of the original theorem, using relative cohomology to analyse the actions of an-annular atoroidal groups on profinite trees.
There has been much recent interest into those properties of a 3-manifold determined by the profinite completion of its fundamental group. In this paper we give readily computable criteria specifying precisely when two orientable graph manifold groups have isomorphic profinite completions. Our results also distinguish graph manifolds among the class of all 3-manifolds and give information about the structure of totally hyperbolic manifolds, and give control over the pro-$p$ completion of certain graph manifold groups.
It is well-known that there are 19 classes of geometries for 4-dimensional manifolds in the sense of Thurston. We could ask that to what extent the geometric information is revealed by the profinite completion of the fundamental group of a closed smooth geometric 4-manifold. In this paper, we show that the geometry of a closed orientable 4-manifold in the sense of Thurston could be detected by the profinite completion of its fundamental group except when the geometry is $ \mathbb{H}^{4}$, $\mathbb{H}^{2}_{\mathbb{C}}$ or $\mathbb{H}^2 \times \mathbb{H}^2$. Moreover, despite the fact that not e…
An interesting question is whether two 3-manifolds can be distinguished by computing and comparing their collections of finite covers; more precisely, by the profinite completions of their fundamental groups. In this paper, we solve this question completely for closed orientable Seifert fibre spaces. In particular, all Seifert fibre spaces are distinguished from each other by their profinite completions apart from some previously-known examples due to Hempel. We also characterize when bounded Seifert fibre space groups have isomorphic profinite completions, given some conditions on the boundar…
Using quantum representations of mapping class groups we prove that profinite completions of Burnside-type surface group quotients are not virtually prosolvable, in general. Further, we construct infinitely many finite simple characteristic quotients of surface groups.
We completely describe the finitely generated pro-$p$ subgroups of the profinite completion of the fundamental group of an arbitrary $3$-manifold. We also prove a pro-$p$ analogue of the main theorem of Bass--Serre theory for finitely generated pro-$p$ groups.
We study several properties of the completed group ring $\widehat{\mathbb{Z}}[[t^{\widehat{\mathbb{Z}}}]]$ and the completed Alexander modules of knots. Then we prove that if the profinite completions of the groups of two knots $J$ and $K$ are isomorphic, then their Alexander polynomials $Δ_J(t)$ and $Δ_K(t)$ coincide.
The profinite completion of the fundamental group of a closed, orientable $3$-manifold determines the Kneser--Milnor decomposition. If $M$ is irreducible, then the profinite completion determines the Jaco--Shalen--Johannson decomposition of $M$.
In this paper we prove that RAAGs are distinguished from each other by their pro-$p$ completions for any choice of prime $p$, and that RACGs are distinguished from each other by their pro-2 completions. We also give a new proof that hyperbolic virtually special groups are good in the sense of Serre. Furthermore we give an example of a property of the underlying graph of a RAAG that translates to a property of the profinite completion.
Using conjugation of Shimura varieties, we produce nonisomorphic, cocompact, torsion-free lattices in $\mathrm{PU}(n,1)$ with isomorphic profinite completions for all $n \ge 2$. This disproves a conjecture of D. Kazhdan and gives the first examples nonisomorphic lattices in a semisimple Lie group of real rank one with isomorphic profinite completions, answering two questions of A. Reid.
Surface groups are determined among limit groups by their profinite completions. As a corollary, the set of surface words in a free group is closed in the profinite topology.
In this paper we define and develop the theory of the cohomology of a profinite group relative to a collection of closed subgroups. Having made the relevant definitions we establish a robust theory of cup products and use this theory to define profinite Poincaré duality pairs. We use the theory of groups acting on profinite trees to give Mayer-Vietoris sequences, and apply this to give results concerning decompositions of 3-manifold groups. Finally we discuss the relationship between discrete duality pairs and profinite duality pairs, culminating in the result that profinite completion of the …
We prove that the profinite completion of the fundamental group of a compact 3-manifold $M$ satisfies a Tits alternative: if a closed subgroup $H$ does not contain a free pro-$p$ subgroup for any $p$, then $H$ is virtually soluble, and furthermore of a very particular form. In particular, the profinite completion of the fundamental group of a closed, hyperbolic 3-manifold does not contain a subgroup isomorphic to $\hat{\mathbb{Z}}^2$. This gives a profinite characterization of hyperbolicity among irreducible 3-manifolds. We also characterize Seifert fibred 3-manifolds as precisely those for wh…
It is proved that the profinite completion of the mapping class group Mod (g,n) of a surface of genus g with n boundary components is isomorphic to such of the arithmetic group GL(6g-6+2n, Z). We establish a relation between the normal subgroups of Mod (g,n) and the absolute Galois group G(K) of a number field K. Using the Tits alternative, we prove the Shafarevich Conjecture saying that the group G(Q^ab) of the maximal abelian extension of the field of rationals is isomorphic to a free profinite group.
If $M$ is a compact 3-manifold whose first betti number is 1, and $N$ is a compact 3-manifold such that $π_1N$ and $π_1M$ have the same finite quotients, then $M$ fibres over the circle if and only if $N$ does. We prove that groups of the form $F_2\rtimes\mathbb{Z}$ are distinguished from one another by their profinite completions. Thus, regardless of betti number, if $M$ and $N$ are punctured torus bundles over the circle and $M$ is not homeomorphic to $N$, then there is a finite group $G$ such that one of $π_1M$ and $π_1N$ maps onto $G$ and the other does not.
